Optimal. Leaf size=164 \[ \frac {x^2 \sqrt {a^2+2 a b x+b^2 x^2} (a A e+a B d+A b d)}{2 (a+b x)}+\frac {x^3 \sqrt {a^2+2 a b x+b^2 x^2} (a B e+A b e+b B d)}{3 (a+b x)}+\frac {a A d x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B e x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 77} \begin {gather*} \frac {x^3 \sqrt {a^2+2 a b x+b^2 x^2} (a B e+A b e+b B d)}{3 (a+b x)}+\frac {x^2 \sqrt {a^2+2 a b x+b^2 x^2} (a A e+a B d+A b d)}{2 (a+b x)}+\frac {a A d x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B e x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (A+B x) (d+e x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b d+b (A b d+a B d+a A e) x+b (b B d+A b e+a B e) x^2+b^2 B e x^3\right ) \, dx}{a b+b^2 x}\\ &=\frac {a A d x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {(A b d+a B d+a A e) x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {(b B d+A b e+a B e) x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b B e x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.45 \begin {gather*} \frac {x \sqrt {(a+b x)^2} (2 a (3 A (2 d+e x)+B x (3 d+2 e x))+b x (A (6 d+4 e x)+B x (4 d+3 e x)))}{12 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.12, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 52, normalized size = 0.32 \begin {gather*} \frac {1}{4} \, B b e x^{4} + A a d x + \frac {1}{3} \, {\left (B b d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 114, normalized size = 0.70 \begin {gather*} \frac {1}{4} \, B b x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B b d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B a x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A b x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a d x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.46 \begin {gather*} \frac {\left (3 B b e \,x^{3}+4 x^{2} A b e +4 x^{2} a B e +4 b B d \,x^{2}+6 x a A e +6 x A b d +6 x a B d +12 A a d \right ) \sqrt {\left (b x +a \right )^{2}}\, x}{12 b x +12 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 254, normalized size = 1.55 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A d x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} e x}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a d}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3} e}{2 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (B d + A e\right )} a x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B e x}{4 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (B d + A e\right )} a^{2}}{2 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a e}{12 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (B d + A e\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 223, normalized size = 1.36 \begin {gather*} \frac {B\,d\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^4}+\frac {B\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}+\frac {A\,\left (a+b\,x\right )\,\left (3\,b\,d-a\,e+2\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,b^2}-\frac {B\,a^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}-\frac {5\,B\,a\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{96\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 63, normalized size = 0.38 \begin {gather*} A a d x + \frac {B b e x^{4}}{4} + x^{3} \left (\frac {A b e}{3} + \frac {B a e}{3} + \frac {B b d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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